TSTP Solution File: LAT213-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT213-1 : TPTP v8.1.2. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:47 EDT 2023
% Result : Unsatisfiable 28.42s 4.01s
% Output : Proof 28.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LAT213-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n017.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu Aug 24 09:21:54 EDT 2023
% 0.14/0.35 % CPUTime :
% 28.42/4.01 Command-line arguments: --no-flatten-goal
% 28.42/4.01
% 28.42/4.01 % SZS status Unsatisfiable
% 28.42/4.01
% 28.91/4.06 % SZS output start Proof
% 28.91/4.06 Take the following subset of the input axioms:
% 28.91/4.06 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 28.91/4.06 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 28.91/4.06 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 28.91/4.06 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 28.91/4.06 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 28.91/4.06 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 28.91/4.06 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 28.91/4.06 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 28.91/4.06 fof(equation_H76, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, join(Y2, U))))=meet(X2, join(Y2, meet(Z2, join(U, meet(X2, Y2)))))).
% 28.91/4.06 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 28.91/4.06 fof(idempotence_of_meet, axiom, ![X2]: meet(X2, X2)=X2).
% 28.91/4.06 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 28.91/4.06 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 28.91/4.06
% 28.91/4.06 Now clausify the problem and encode Horn clauses using encoding 3 of
% 28.91/4.06 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 28.91/4.06 We repeatedly replace C & s=t => u=v by the two clauses:
% 28.91/4.06 fresh(y, y, x1...xn) = u
% 28.91/4.06 C => fresh(s, t, x1...xn) = v
% 28.91/4.06 where fresh is a fresh function symbol and x1..xn are the free
% 28.91/4.06 variables of u and v.
% 28.91/4.06 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 28.91/4.06 input problem has no model of domain size 1).
% 28.91/4.06
% 28.91/4.06 The encoding turns the above axioms into the following unit equations and goals:
% 28.91/4.06
% 28.91/4.06 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 28.91/4.06 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 28.91/4.06 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 28.91/4.06 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 28.91/4.06 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 28.91/4.06 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 28.91/4.06 Axiom 7 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 28.91/4.06 Axiom 8 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 28.91/4.06 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 28.91/4.06 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 28.91/4.06 Axiom 11 (absorption1): meet(X, join(X, Y)) = X.
% 28.91/4.06 Axiom 12 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 28.91/4.06 Axiom 13 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 28.91/4.06 Axiom 14 (equation_H76): meet(X, join(Y, meet(Z, join(Y, W)))) = meet(X, join(Y, meet(Z, join(W, meet(X, Y))))).
% 28.91/4.06
% 28.91/4.06 Lemma 15: complement(complement(X)) = X.
% 28.91/4.06 Proof:
% 28.91/4.06 complement(complement(X))
% 28.91/4.06 = { by axiom 8 (meet_join_complement) R->L }
% 28.91/4.06 fresh2(one, one, complement(X), X)
% 28.91/4.06 = { by axiom 5 (complement_join) R->L }
% 28.91/4.06 fresh2(join(X, complement(X)), one, complement(X), X)
% 28.91/4.06 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.06 fresh2(join(complement(X), X), one, complement(X), X)
% 28.91/4.06 = { by axiom 13 (meet_join_complement) }
% 28.91/4.06 fresh(meet(complement(X), X), zero, complement(X), X)
% 28.91/4.06 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.06 fresh(meet(X, complement(X)), zero, complement(X), X)
% 28.91/4.06 = { by axiom 6 (complement_meet) }
% 28.91/4.06 fresh(zero, zero, complement(X), X)
% 28.91/4.06 = { by axiom 7 (meet_join_complement) }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 16: meet(X, one) = X.
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, one)
% 28.91/4.07 = { by axiom 5 (complement_join) R->L }
% 28.91/4.07 meet(X, join(X, complement(X)))
% 28.91/4.07 = { by axiom 11 (absorption1) }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 17: join(X, zero) = X.
% 28.91/4.07 Proof:
% 28.91/4.07 join(X, zero)
% 28.91/4.07 = { by axiom 6 (complement_meet) R->L }
% 28.91/4.07 join(X, meet(X, complement(X)))
% 28.91/4.07 = { by axiom 9 (absorption2) }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 18: meet(X, zero) = zero.
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, zero)
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.07 meet(zero, X)
% 28.91/4.07 = { by lemma 17 R->L }
% 28.91/4.07 join(meet(zero, X), zero)
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.07 join(zero, meet(zero, X))
% 28.91/4.07 = { by axiom 9 (absorption2) }
% 28.91/4.07 zero
% 28.91/4.07
% 28.91/4.07 Lemma 19: join(X, one) = one.
% 28.91/4.07 Proof:
% 28.91/4.07 join(X, one)
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 join(one, X)
% 28.91/4.07 = { by lemma 16 R->L }
% 28.91/4.07 join(one, meet(X, one))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.07 join(one, meet(one, X))
% 28.91/4.07 = { by axiom 9 (absorption2) }
% 28.91/4.07 one
% 28.91/4.07
% 28.91/4.07 Lemma 20: meet(X, join(Y, X)) = X.
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, join(Y, X))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 meet(X, join(X, Y))
% 28.91/4.07 = { by axiom 11 (absorption1) }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 21: join(X, meet(Y, X)) = X.
% 28.91/4.07 Proof:
% 28.91/4.07 join(X, meet(Y, X))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.07 join(X, meet(X, Y))
% 28.91/4.07 = { by axiom 9 (absorption2) }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 22: join(X, join(Y, X)) = join(X, Y).
% 28.91/4.07 Proof:
% 28.91/4.07 join(X, join(Y, X))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 join(X, join(X, Y))
% 28.91/4.07 = { by axiom 10 (associativity_of_join) R->L }
% 28.91/4.07 join(join(X, X), Y)
% 28.91/4.07 = { by axiom 1 (idempotence_of_join) }
% 28.91/4.07 join(X, Y)
% 28.91/4.07
% 28.91/4.07 Lemma 23: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 28.91/4.07 Proof:
% 28.91/4.07 join(Y, join(X, Z))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 join(join(X, Z), Y)
% 28.91/4.07 = { by axiom 10 (associativity_of_join) }
% 28.91/4.07 join(X, join(Z, Y))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.07 join(X, join(Y, Z))
% 28.91/4.07
% 28.91/4.07 Lemma 24: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, join(Y, meet(Z, join(X, Y))))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 meet(X, join(Y, meet(Z, join(Y, X))))
% 28.91/4.07 = { by axiom 14 (equation_H76) }
% 28.91/4.07 meet(X, join(Y, meet(Z, join(X, meet(X, Y)))))
% 28.91/4.07 = { by axiom 9 (absorption2) }
% 28.91/4.07 meet(X, join(Y, meet(Z, X)))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.07 meet(X, join(Y, meet(X, Z)))
% 28.91/4.07
% 28.91/4.07 Lemma 25: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, join(complement(X), meet(X, Y)))
% 28.91/4.07 = { by lemma 24 R->L }
% 28.91/4.07 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 28.91/4.07 = { by axiom 5 (complement_join) }
% 28.91/4.07 meet(X, join(complement(X), meet(Y, one)))
% 28.91/4.07 = { by lemma 16 }
% 28.91/4.07 meet(X, join(complement(X), Y))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.07 meet(X, join(Y, complement(X)))
% 28.91/4.07
% 28.91/4.07 Lemma 26: join(X, join(Y, complement(X))) = one.
% 28.91/4.07 Proof:
% 28.91/4.07 join(X, join(Y, complement(X)))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 join(X, join(complement(X), Y))
% 28.91/4.07 = { by axiom 10 (associativity_of_join) R->L }
% 28.91/4.07 join(join(X, complement(X)), Y)
% 28.91/4.07 = { by axiom 5 (complement_join) }
% 28.91/4.07 join(one, Y)
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 join(Y, one)
% 28.91/4.07 = { by lemma 19 }
% 28.91/4.07 one
% 28.91/4.07
% 28.91/4.07 Lemma 27: fresh(meet(complement(X), join(X, Y)), zero, complement(X), join(X, Y)) = X.
% 28.91/4.07 Proof:
% 28.91/4.07 fresh(meet(complement(X), join(X, Y)), zero, complement(X), join(X, Y))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 fresh(meet(complement(X), join(Y, X)), zero, complement(X), join(X, Y))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 fresh(meet(complement(X), join(Y, X)), zero, complement(X), join(Y, X))
% 28.91/4.07 = { by lemma 15 R->L }
% 28.91/4.07 fresh(meet(complement(X), join(Y, X)), zero, complement(X), join(Y, complement(complement(X))))
% 28.91/4.07 = { by lemma 15 R->L }
% 28.91/4.07 fresh(meet(complement(X), join(Y, complement(complement(X)))), zero, complement(X), join(Y, complement(complement(X))))
% 28.91/4.07 = { by axiom 13 (meet_join_complement) R->L }
% 28.91/4.07 fresh2(join(complement(X), join(Y, complement(complement(X)))), one, complement(X), join(Y, complement(complement(X))))
% 28.91/4.07 = { by lemma 26 }
% 28.91/4.07 fresh2(one, one, complement(X), join(Y, complement(complement(X))))
% 28.91/4.07 = { by axiom 8 (meet_join_complement) }
% 28.91/4.07 complement(complement(X))
% 28.91/4.07 = { by lemma 15 }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 28: meet(complement(X), join(X, meet(complement(X), Y))) = meet(complement(X), join(X, Y)).
% 28.91/4.07 Proof:
% 28.91/4.07 meet(complement(X), join(X, meet(complement(X), Y)))
% 28.91/4.07 = { by lemma 15 R->L }
% 28.91/4.07 meet(complement(X), join(complement(complement(X)), meet(complement(X), Y)))
% 28.91/4.07 = { by lemma 25 }
% 28.91/4.07 meet(complement(X), join(Y, complement(complement(X))))
% 28.91/4.07 = { by lemma 15 }
% 28.91/4.07 meet(complement(X), join(Y, X))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.07 meet(complement(X), join(X, Y))
% 28.91/4.07
% 28.91/4.07 Lemma 29: meet(X, complement(meet(Y, complement(X)))) = X.
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, complement(meet(Y, complement(X))))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.07 meet(X, complement(meet(complement(X), Y)))
% 28.91/4.07 = { by lemma 15 R->L }
% 28.91/4.07 meet(complement(complement(X)), complement(meet(complement(X), Y)))
% 28.91/4.07 = { by axiom 7 (meet_join_complement) R->L }
% 28.91/4.07 fresh(zero, zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 18 R->L }
% 28.91/4.07 fresh(meet(complement(meet(complement(X), Y)), zero), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.07 fresh(meet(zero, complement(meet(complement(X), Y))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 6 (complement_meet) R->L }
% 28.91/4.07 fresh(meet(meet(complement(X), complement(complement(X))), complement(meet(complement(X), Y))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 12 (associativity_of_meet) }
% 28.91/4.07 fresh(meet(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 13 (meet_join_complement) R->L }
% 28.91/4.07 fresh2(join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 21 R->L }
% 28.91/4.07 fresh2(join(join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), meet(complement(complement(X)), join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 28 }
% 28.91/4.07 fresh2(join(join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 10 (associativity_of_join) }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), complement(meet(complement(X), Y))), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(meet(complement(X), Y)), complement(complement(X))), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(meet(complement(X), Y)), complement(complement(X))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(complement(X)), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 20 R->L }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(complement(X)), meet(complement(meet(complement(X), Y)), join(complement(X), complement(meet(complement(X), Y))))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(meet(complement(meet(complement(X), Y)), join(complement(X), complement(meet(complement(X), Y)))), complement(complement(X))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 12 (associativity_of_meet) }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(meet(complement(X), Y)), meet(join(complement(X), complement(meet(complement(X), Y))), complement(complement(X)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.07 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(meet(complement(X), Y)), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y))))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 21 }
% 28.91/4.07 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 9 (absorption2) R->L }
% 28.91/4.07 fresh2(join(complement(X), meet(complement(complement(X)), join(join(complement(X), meet(complement(X), Y)), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 10 (associativity_of_join) }
% 28.91/4.07 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), join(meet(complement(X), Y), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 5 (complement_join) }
% 28.91/4.07 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), one))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 19 }
% 28.91/4.07 fresh2(join(complement(X), meet(complement(complement(X)), one)), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by lemma 16 }
% 28.91/4.07 fresh2(join(complement(X), complement(complement(X))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 5 (complement_join) }
% 28.91/4.07 fresh2(one, one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.91/4.07 = { by axiom 8 (meet_join_complement) }
% 28.91/4.07 complement(complement(X))
% 28.91/4.07 = { by lemma 15 }
% 28.91/4.07 X
% 28.91/4.07
% 28.91/4.07 Lemma 30: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 28.91/4.07 Proof:
% 28.91/4.07 meet(X, complement(meet(X, Y)))
% 28.91/4.07 = { by lemma 15 R->L }
% 28.91/4.07 meet(X, complement(meet(X, complement(complement(Y)))))
% 28.91/4.07 = { by lemma 20 R->L }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y)))))))
% 28.91/4.07 = { by axiom 7 (meet_join_complement) R->L }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(zero, zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.07 = { by axiom 6 (complement_meet) R->L }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), complement(complement(complement(Y)))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.07 = { by lemma 17 R->L }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), zero)), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.07 = { by axiom 6 (complement_meet) R->L }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(meet(complement(complement(Y)), X), complement(meet(complement(complement(Y)), X))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.07 = { by axiom 12 (associativity_of_meet) }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(complement(complement(Y)), meet(X, complement(meet(complement(complement(Y)), X)))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.07 = { by lemma 25 }
% 28.91/4.07 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(meet(X, complement(meet(complement(complement(Y)), X))), complement(complement(complement(Y))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.07 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.08 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(X, complement(meet(complement(complement(Y)), X))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.08 = { by lemma 15 }
% 28.91/4.08 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, complement(meet(complement(complement(Y)), X))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.91/4.08 = { by lemma 27 }
% 28.91/4.08 meet(meet(X, complement(meet(X, complement(complement(Y))))), complement(Y))
% 28.91/4.08 = { by axiom 12 (associativity_of_meet) }
% 28.91/4.08 meet(X, meet(complement(meet(X, complement(complement(Y)))), complement(Y)))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(X, meet(complement(Y), complement(meet(X, complement(complement(Y))))))
% 28.91/4.08 = { by lemma 29 }
% 28.91/4.08 meet(X, complement(Y))
% 28.91/4.08
% 28.91/4.08 Lemma 31: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 28.91/4.08 Proof:
% 28.91/4.08 meet(X, complement(meet(Y, X)))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 meet(X, complement(meet(X, Y)))
% 28.91/4.08 = { by lemma 30 }
% 28.91/4.08 meet(X, complement(Y))
% 28.91/4.08
% 28.91/4.08 Lemma 32: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 28.91/4.08 Proof:
% 28.91/4.08 meet(complement(X), join(X, Y))
% 28.91/4.08 = { by lemma 15 R->L }
% 28.91/4.08 meet(complement(X), join(X, complement(complement(Y))))
% 28.91/4.08 = { by lemma 28 R->L }
% 28.91/4.08 meet(complement(X), join(X, meet(complement(X), complement(complement(Y)))))
% 28.91/4.08 = { by lemma 31 R->L }
% 28.91/4.08 meet(complement(X), join(X, meet(complement(X), complement(meet(complement(Y), complement(X))))))
% 28.91/4.08 = { by lemma 28 }
% 28.91/4.08 meet(complement(X), join(X, complement(meet(complement(Y), complement(X)))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(complement(X), join(complement(meet(complement(Y), complement(X))), X))
% 28.91/4.08 = { by lemma 29 R->L }
% 28.91/4.08 meet(complement(X), join(complement(meet(complement(Y), complement(X))), meet(X, complement(meet(complement(Y), complement(X))))))
% 28.91/4.08 = { by lemma 21 }
% 28.91/4.08 meet(complement(X), complement(meet(complement(Y), complement(X))))
% 28.91/4.08 = { by lemma 31 }
% 28.91/4.08 meet(complement(X), complement(complement(Y)))
% 28.91/4.08 = { by lemma 15 }
% 28.91/4.08 meet(complement(X), Y)
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(Y, complement(X))
% 28.91/4.08
% 28.91/4.08 Lemma 33: meet(X, complement(join(Y, complement(X)))) = complement(join(Y, complement(X))).
% 28.91/4.08 Proof:
% 28.91/4.08 meet(X, complement(join(Y, complement(X))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), X)
% 28.91/4.08 = { by lemma 27 R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, complement(join(Y, complement(X))))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, complement(join(complement(X), Y)))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by lemma 15 R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(complement(X)), complement(join(complement(X), Y)))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(join(complement(X), Y)), complement(complement(X)))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by lemma 25 R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(complement(X)), meet(complement(X), complement(join(complement(X), Y))))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 11 (absorption1) R->L }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(complement(X)), meet(meet(complement(X), join(complement(X), Y)), complement(join(complement(X), Y))))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 12 (associativity_of_meet) }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(complement(X)), meet(complement(X), meet(join(complement(X), Y), complement(join(complement(X), Y)))))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 6 (complement_meet) }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(complement(X)), meet(complement(X), zero))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by lemma 18 }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(complement(X)), zero)), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by lemma 17 }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(meet(complement(X), complement(complement(X))), zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 6 (complement_meet) }
% 28.91/4.08 meet(complement(join(Y, complement(X))), fresh(zero, zero, complement(X), join(X, complement(join(Y, complement(X))))))
% 28.91/4.08 = { by axiom 7 (meet_join_complement) }
% 28.91/4.08 meet(complement(join(Y, complement(X))), join(X, complement(join(Y, complement(X)))))
% 28.91/4.08 = { by lemma 20 }
% 28.91/4.08 complement(join(Y, complement(X)))
% 28.91/4.08
% 28.91/4.08 Lemma 34: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 28.91/4.08 Proof:
% 28.91/4.08 join(complement(X), meet(X, Y))
% 28.91/4.08 = { by lemma 15 R->L }
% 28.91/4.08 complement(complement(join(complement(X), meet(X, Y))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 complement(complement(join(complement(X), meet(Y, X))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 complement(complement(join(meet(Y, X), complement(X))))
% 28.91/4.08 = { by lemma 33 R->L }
% 28.91/4.08 complement(meet(X, complement(join(meet(Y, X), complement(X)))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.08 complement(meet(X, complement(join(complement(X), meet(Y, X)))))
% 28.91/4.08 = { by lemma 30 R->L }
% 28.91/4.08 complement(meet(X, complement(meet(X, join(complement(X), meet(Y, X))))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 complement(meet(X, complement(meet(X, join(complement(X), meet(X, Y))))))
% 28.91/4.08 = { by lemma 25 }
% 28.91/4.08 complement(meet(X, complement(meet(X, join(Y, complement(X))))))
% 28.91/4.08 = { by lemma 30 }
% 28.91/4.08 complement(meet(X, complement(join(Y, complement(X)))))
% 28.91/4.08 = { by lemma 33 }
% 28.91/4.08 complement(complement(join(Y, complement(X))))
% 28.91/4.08 = { by lemma 15 }
% 28.91/4.08 join(Y, complement(X))
% 28.91/4.08
% 28.91/4.08 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 28.91/4.08 Proof:
% 28.91/4.08 meet(a, join(b, c))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(a, join(c, b))
% 28.91/4.08 = { by lemma 15 R->L }
% 28.91/4.08 meet(complement(complement(a)), join(c, b))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 meet(join(c, b), complement(complement(a)))
% 28.91/4.08 = { by lemma 32 R->L }
% 28.91/4.08 meet(complement(complement(a)), join(complement(a), join(c, b)))
% 28.91/4.08 = { by lemma 23 }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(complement(a), b)))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(b, complement(a))))
% 28.91/4.08 = { by lemma 15 R->L }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(b, complement(complement(complement(a))))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(complement(complement(complement(a))), b)))
% 28.91/4.08 = { by lemma 16 R->L }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(complement(complement(complement(a))), meet(b, one))))
% 28.91/4.08 = { by axiom 10 (associativity_of_join) R->L }
% 28.91/4.08 meet(complement(complement(a)), join(join(c, complement(complement(complement(a)))), meet(b, one)))
% 28.91/4.08 = { by lemma 26 R->L }
% 28.91/4.08 meet(complement(complement(a)), join(join(c, complement(complement(complement(a)))), meet(b, join(complement(complement(a)), join(c, complement(complement(complement(a))))))))
% 28.91/4.08 = { by lemma 24 }
% 28.91/4.08 meet(complement(complement(a)), join(join(c, complement(complement(complement(a)))), meet(complement(complement(a)), b)))
% 28.91/4.08 = { by axiom 10 (associativity_of_join) }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(complement(complement(complement(a))), meet(complement(complement(a)), b))))
% 28.91/4.08 = { by lemma 15 }
% 28.91/4.08 meet(complement(complement(a)), join(c, join(complement(a), meet(complement(complement(a)), b))))
% 28.91/4.08 = { by lemma 23 R->L }
% 28.91/4.08 meet(complement(complement(a)), join(complement(a), join(c, meet(complement(complement(a)), b))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(complement(complement(a)), join(complement(a), join(c, meet(b, complement(complement(a))))))
% 28.91/4.08 = { by lemma 32 }
% 28.91/4.08 meet(join(c, meet(b, complement(complement(a)))), complement(complement(a)))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(complement(complement(a)), join(c, meet(b, complement(complement(a)))))
% 28.91/4.08 = { by lemma 15 }
% 28.91/4.08 meet(a, join(c, meet(b, complement(complement(a)))))
% 28.91/4.08 = { by lemma 15 }
% 28.91/4.08 meet(a, join(c, meet(b, a)))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(a, join(c, meet(a, b)))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), a)
% 28.91/4.08 = { by axiom 3 (idempotence_of_meet) R->L }
% 28.91/4.08 meet(meet(join(c, meet(a, b)), join(c, meet(a, b))), a)
% 28.91/4.08 = { by axiom 12 (associativity_of_meet) }
% 28.91/4.08 meet(join(c, meet(a, b)), meet(join(c, meet(a, b)), a))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(join(c, meet(a, b)), meet(a, join(c, meet(a, b))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), meet(a, join(c, meet(b, a))))
% 28.91/4.08 = { by lemma 21 R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, join(c, meet(b, a))), meet(b, meet(a, join(c, meet(b, a))))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, join(c, meet(b, a))), meet(b, meet(a, join(meet(b, a), c)))))
% 28.91/4.08 = { by axiom 12 (associativity_of_meet) R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, join(c, meet(b, a))), meet(meet(b, a), join(meet(b, a), c))))
% 28.91/4.08 = { by axiom 11 (absorption1) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, join(c, meet(b, a))), meet(b, a)))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(b, a), meet(a, join(c, meet(b, a)))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(b, a), meet(a, join(c, meet(a, b)))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, b), meet(a, join(c, meet(a, b)))))
% 28.91/4.08 = { by lemma 22 R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, b), join(meet(a, join(c, meet(a, b))), meet(a, b))))
% 28.91/4.08 = { by lemma 15 R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(meet(a, b), join(meet(a, join(c, meet(a, b))), complement(complement(meet(a, b))))))
% 28.91/4.08 = { by axiom 10 (associativity_of_join) R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(join(meet(a, b), meet(a, join(c, meet(a, b)))), complement(complement(meet(a, b)))))
% 28.91/4.08 = { by lemma 34 R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(complement(complement(meet(a, b))), meet(complement(meet(a, b)), join(meet(a, b), meet(a, join(c, meet(a, b)))))))
% 28.91/4.08 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.08 meet(join(c, meet(a, b)), join(complement(complement(meet(a, b))), meet(complement(meet(a, b)), join(meet(a, b), meet(a, join(meet(a, b), c))))))
% 28.91/4.08 = { by axiom 14 (equation_H76) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(complement(complement(meet(a, b))), meet(complement(meet(a, b)), join(meet(a, b), meet(a, join(c, meet(complement(meet(a, b)), meet(a, b))))))))
% 28.91/4.08 = { by axiom 4 (commutativity_of_meet) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(complement(complement(meet(a, b))), meet(complement(meet(a, b)), join(meet(a, b), meet(a, join(c, meet(meet(a, b), complement(meet(a, b)))))))))
% 28.91/4.08 = { by axiom 6 (complement_meet) }
% 28.91/4.08 meet(join(c, meet(a, b)), join(complement(complement(meet(a, b))), meet(complement(meet(a, b)), join(meet(a, b), meet(a, join(c, zero))))))
% 28.91/4.08 = { by lemma 17 }
% 28.91/4.08 meet(join(c, meet(a, b)), join(complement(complement(meet(a, b))), meet(complement(meet(a, b)), join(meet(a, b), meet(a, c)))))
% 28.91/4.09 = { by lemma 34 }
% 28.91/4.09 meet(join(c, meet(a, b)), join(join(meet(a, b), meet(a, c)), complement(complement(meet(a, b)))))
% 28.91/4.09 = { by axiom 10 (associativity_of_join) }
% 28.91/4.09 meet(join(c, meet(a, b)), join(meet(a, b), join(meet(a, c), complement(complement(meet(a, b))))))
% 28.91/4.09 = { by lemma 15 }
% 28.91/4.09 meet(join(c, meet(a, b)), join(meet(a, b), join(meet(a, c), meet(a, b))))
% 28.91/4.09 = { by lemma 22 }
% 28.91/4.09 meet(join(c, meet(a, b)), join(meet(a, b), meet(a, c)))
% 28.91/4.09 = { by axiom 4 (commutativity_of_meet) R->L }
% 28.91/4.09 meet(join(meet(a, b), meet(a, c)), join(c, meet(a, b)))
% 28.91/4.09 = { by axiom 2 (commutativity_of_join) R->L }
% 28.91/4.09 meet(join(meet(a, b), meet(a, c)), join(meet(a, b), c))
% 28.91/4.09 = { by lemma 21 R->L }
% 28.91/4.09 meet(join(meet(a, b), meet(a, c)), join(meet(a, b), join(c, meet(a, c))))
% 28.91/4.09 = { by lemma 23 R->L }
% 28.91/4.09 meet(join(meet(a, b), meet(a, c)), join(c, join(meet(a, b), meet(a, c))))
% 28.91/4.09 = { by lemma 20 }
% 28.91/4.09 join(meet(a, b), meet(a, c))
% 28.91/4.09 % SZS output end Proof
% 28.91/4.09
% 28.91/4.09 RESULT: Unsatisfiable (the axioms are contradictory).
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